A291717 Triangle T(m,k) read by rows, where T(m,k) is the number of ways in which 1 <= k <= m positions can be picked in an m X m square grid such that the picked positions have a central symmetry.

1, 4, 6, 9, 36, 8, 16, 120, 24, 168, 25, 300, 72, 714, 178, 36, 630, 144, 2273, 464, 6576, 49, 1176, 288, 5932, 1476, 24288, 6404, 64, 2016, 480, 13536, 3040, 74560, 15680, 341320, 81, 3240, 800, 27860, 6940, 197600, 50860, 1170466, 314862

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Example

A configuration of 6 picked points from a 7 X 7 grid with a central (point) symmetry w.r.t. point #, but no line (mirror) symmetry and thus only contributing to T(7,6)=a(27), but not to A291718(27), would be:
  o o o X o o o
  o o o o o o o
  o o o o X o o
  o X # X o o o
  X o o o o o o
  o o o o o o o
  o X o o o o o
.
Triangle begins:
   1;
   4,    6;
   9,   36,   8;
  16,  120,  24,   168;
  25,  300,  72,   714,  178;
  36,  630, 144,  2273,  464,  6576;
  49, 1176, 288,  5932, 1476, 24288,  6404;
  64, 2016, 480, 13536, 3040, 74560, 15680, 341320;

Mathematica

decentralize[v_] := 2*Total[v] - Last[v];
T[n_, k_] := decentralize[ Table[ decentralize[ Table[ If[EvenQ[k] || OddQ[a*b], Binomial[ Quotient[a*b, 2], Quotient[k, 2]], 0], {b, 1, n}]], {a, 1, n}]];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)

Prog

(PARI)
decentralize(v) = 2*vecsum(v) - v[length(v)];
T(n, k) = decentralize(vector(n, a, decentralize(vector(n, b, if(k%2==0||a*b%2==1, binomial(a*b\2, k\2))))));
for(n=1, 10, for(k=1, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Sep 16 2017

Crossrefs

Cf. A090642, A098485, A098487, A291716, A291718, A292152, A292153, A292154, A292155, A292156.

Sequence in context: A192220 A215477 A292154 * A303699 A292153 A175459

Adjacent sequences: A291714 A291715 A291716 * A291718 A291719 A291720

Keyword

nonn,tabl

Author

Hugo Pfoertner, Sep 08 2017

Extensions

Terms a(37) and beyond from Andrew Howroyd, Sep 16 2017

Status

approved